Well known in the art are internal combustion engines with exhaust systems each of which is provided with a catalytic converter for purifying harmful contaminants, such as hydrocarbons (HC), carbon monoxide (CO) and nitrogen oxides (NO.sub.x) contained in the exhaust gas. According to this well-known exhaust system, secondary air is supplied to a position upstream of the catalytic converter for controlling the secondary air fuel ratio at the stoichiometric air fuel ratio so as to maintain the converting efficiency of the catalytic converter at a high level. (The term "secondary air fuel ratio" used herein, is defined as the total amount of air, including secondary air, supplied to the intake system of the engine and supplied to a position between the cylinders of the engine and the entrance of the catalytic converter, to the total amount of fuel.)
The above-mentioned converting efficiency depends on the activating temperature of the catalytic converter as well as the above-mentioned secondary air fuel ratio. As a result, it is necessary to increase the temperature of the catalytic converter to the predetermined high temperature immediately after the engine is started. On the other hand, when the temperature of the catalytic converter is increased to a temperature which is too high, the converting efficiency of the catalytic converter may be decreased and the vessel which contains the catalyzer may be thermally damaged. Some attempts have been made to facilitate the warming of the catalytic converter when the engine is cold and to eliminate thermal damage of the catalytic converter after the engine has warmed up, by using two catalytic converters disposed in series within the exhaust system of an engine. By utilizing this apparatus when the engine is cold, secondary air is supplied to an exhaust port or to a position adjacent to the exhaust manifold of the engine, which exhaust port or position is located upstream of the first catalytic converter, for facilitating the conversion of exhaust gas by the first catalytic converter and for increasing the temperature of the first catalytic converter to a predetermined level. However, if secondary air is continued to be supplied after the temperature of the first catalytic converter has reached a predetermined level, the first catalytic converter may be thermally damaged by heat generated by the converting reaction of the first catalytic converter and by hot exhaust gas. Accordingly, in the conventional apparatus, secondary air is stopped from being supplied to the first catalytic converter after the temperature of the first catalytic converter has reached a predetermined level so as to prevent the first catalytic converter from thermal damage.
However, as mentioned above, in the conventional apparatus, after the temperature of the first catalytic converter has reached a predetermined level, no secondary air is supplied to the first catalytic converter for preventing the occurrence of thermal damage in the first catalytic converter. As a result, the air in the first catalytic converter becomes insufficient for producing the desirable converting efficiency of the first catalytic converter. In other words, the converting efficiency of the first catalytic converter is not utilized sufficiently.